Integrability of orthogonal projections, and applications to Furstenberg sets

Let G(d,n) be the Grassmannian manifold of n-dimensional subspaces Rd, and let πV:Rd→V orthogonal projection. We prove that if μ is a compactly supported Radon measure on Rd satisfying s-dimensional Frostman condition μ(B(x,r))⩽Crs for all x∈Rd r>0, then∫G(d,n)‖πVμ‖Lp(V)pdγd,n(V)<∞,1⩽p<2d−n−sd−s. The upper bound p sharp, at least, d−1⩽s⩽d, every 012 t⩾1+ϵ small absolute constant ϵ>0. also higher dimensional analogue estimate codimension-1 Furstenberg sets Rd. another corollary method, obtain δ-discretised sum-product (δ,s)-sets. Chen 12<s<1, Guth-Katz-Zahl s⩾0.5151.